We perform regression in the CAltAlt planner to find conformant
plans by starting with the goal belief state and regressing it
non-deterministically over all relevant actions. An action (without
observations) is relevant for regressing a belief state if (i) its
unconditional effect is consistent with every state in the belief
state and (ii) at least one effect consequent contains a literal
that is present in a constituent of the belief state. The first part
of relevance requires that every state in the successor belief state
is actually reachable from the predecessor belief state and the
second ensures that the action helps support the successor.

Following [25], regressing a belief state
over an action
, with conditional effects, involves finding
the execution, causation, and preservation formulas. We define
regression in terms of clausal representation, but it can be
generalized for arbitrary formulas. The regression of a belief state
is a conjunction of the regression of clauses in
.
Formally, the result
of regressing the belief state
over
the action
is defined as:3

Execution formula (
) is the execution
precondition
. This is what must hold in
for
to
have been applicable.

Causation formula (
) for a literal
w.r.t all effects
of an action
is defined as
the weakest formula that must hold in the state before
such that
holds in
. The intuitive meaning is that
already held
in
, or the antecedent
must have held in
to
make
hold in
. Formally
is defined as:

Preservation formula (
) of a literal
w.r.t. all effects
of action
is defined as the
formula that must be true before
such that
is not violated
by any effect
. The intuitive meaning is that the
antecedent of every effect that is inconsistent with
could not
have held in
. Formally
is defined as:

Regression has also been formalized in the MBP planner
[11] as a symbolic pre-image computation of
BDDs [8]. While our formulation is syntactically
different, both approaches compute the same result.